7.9 Other Models of Time-Varying Delays

In this section, models of time-varying delays which are not linearly or quadratically time variant are briefly considered.

7.9.1 Taylor Series Expansion of Range and Delay

The time-varying range R(t) can be expanded in Taylor series, around t₀, with Lagrange residual term:

(7.370)

where R⁽ⁿ⁾(·) denotes the nth-order derivative of R(·), and if t > t₀ and if t < t₀. This expression can be substituted into (c07-mdis-0044) to get an n-order algebric equation in D(t). One of the roots of this equation is the time-varying delay D(t) for the given R(t) (see Section 7.4 for the case n = 2 and ).

Alternatively, by following the approach in (Kelly 1961) and (Kelly and Wishner 1965), the time-varying delay D(t) can be expanded in Taylor series, around t₀, with Lagrange residual term:

(7.371)

where if t > t₀ and if t < t₀.

This approach, when is embedded in additive white Gaussian noise (AWGN), allows to address the joint detection-estimation problem by the generalized ambiguity function (Kelly 1961; Kelly and Wishner 1965).

7.9.2 Periodically Time-Variant Delay

A rotating reflecting object gives rise to a periodically time-variant delay in the received signal. If the object is also moving with constant relative radial speed, then the delay contains also a linearly time-variant term (Chen et al. 2006):

(7.372)

The effect on the received signal is called micro-Doppler. In (Chen et al. 2006), with reference to the complex-envelope received-signal

(7.373)

the “Doppler frequency” is defined as

(7.374)

and the micro-Doppler effect is characterized by time-frequency analysis.

7.9.3 Periodically Time-Variant Carrier Frequency

Let x₀(t) be the complex-envelope of a transmitted signal and let

(7.375)

the complex signal reflected under the narrow-band condition by an oscillating scatterer whose effect is to produce a sinusoidally time-varying time-scale factor. The same mathematical model is obtained if the carrier frequency is not constant but is sinusoidally time-varying due to imperfection of the transmitting oscillator.

The (conjugate) autocorrelation function of x(t) is

(7.376)

where

(7.377)

Let a 2πf_cΔ and ω_m 2πf_m. By using the expansion of a complex exponential in terms of Bessel functions (NIST 2010, eqs. 10.12.1-3) (with z = au and θ = 2πf_mt + ϕ) we have

(7.378)

where J_n(z) is the Bessel function of the first kind of order n (NIST 2010, eqs. 10.2.2, 10.9.1, 10.9.2). The function of two variables ξ(u, t) can be evaluated along the diagonal u = t leading to

(7.379)

The second-order lag product of z(t) is given by

(7.380)

The function r₁(t) is periodic in t. From (7.378) with u = τ we have

(7.381)

Due to the function

(7.382)

the signal x(t) turns out to be GACS with nonlinear lag-dependent cycle frequencies with complicate analytical expression.

A simulation experiment is carried out to estimate the cyclic correlogram of x(t). In the experiment, x₀(t) is a PAM signal with raised cosine pulse with excess bandwidth η = 0.25, stationary white binary modulating sequence, and symbol period T_p = 64T_s, where T_s is the sampling period. Furthermore, f_c = 0.125/T_s, s = 0.99, f_m = 0.0233/T_s, Δ = 0.005, and ϕ = 0.

In Figure 7.19, (top) graph and (bottom) “checkerboard” plot of the magnitude of the cyclic correlogram of x(t), estimated by 2¹³ samples as a function of αT_s and τ/T_s, are reported.

Figure 7.19 Complex PAM signal with sinusoidally-varying carrier frequency. (Top) graph and (bottom) “checkerboard” plot of the magnitude of the cyclic correlogram as a function of αT_s and τ/T_s